Elementary Biometrical Inferences 



525 



of successes and failures by expanding the 

 binomial expression (q + p) N - If a "one" 

 on a die is a success and the die is tossed 

 5 times, the probabilities of ones, 1 one, 

 2 ones, 3 ones, etc., among the 5 tosses are 

 given by successive terms of the expansion 

 of the binomial 



(f+5)' 



In this expression % represents the proba- 

 bility of not obtaining a one on a single 

 trial, }/q the probability of obtaining a one, 

 and the exponent 5 the number of trials. 

 The expansion is shown in Table 1, below. 

 Note that each result is possible, each hav- 

 ing its own exact probability of occurrence. 



PROBLEMS 



A.ll. If you roll a die three times, what is 

 the probability of obtaining (a) three 

 "fours" in succession? (b) "One," 

 "two," and "three" in that order? 



A. 12. If you roll two dice at the same time 

 in a single trial, what is the prob- 

 ability of obtaining a total of eleven? 

 Two? Seven? 



A. 13. What is the chance that a simul- 

 taneous toss of a penny, a nickel, 

 a dime, a quarter, and a half-dollar 

 will fall : 



(a) All heads or all tails? 



(b) 3 heads and 2 tails? 



A. 14. What is the exact probability (using 

 an unbiased penny) of a run of tosses 

 which: 



(a) Starts with 2 heads and ends 

 with 3 tails? 



(b) Has 4 successive heads? 



(c) Has 5 successive tails? 



A. 15. What is the exact probability of 10 

 successes, if p = i^ and N = 1.5? 



A. 16. How often will you expect to obtain 

 less than 3 successes if p = 34 and 

 N = 5? 



A. 17. You have just etherized Drosophila 

 which are the progeny of a cross be- 

 tween ci + ci and ci + ci. What is the 

 probability that there is only 1 ci ci 

 fly among the first 3 flies chosen at 

 random? Among the first 5 flies 

 chosen at random? 



A. 18. Following independent segregation of 

 Aa Bb Cc, an ascus is formed. What 

 is the probability that if two asco- 

 spores are chosen at random they will 

 be A B C? a be? Either A B C or 

 a b c? 



A. 19. An albino (aa) man of blood type 

 MN marries a heterozygote for al- 

 binism (Aa) also of MN blood type. 

 They plan to have 4 children. If 

 you assume independent segregation, 

 what is the exact probability that 

 they will have: 



(a) No albinos? 



(b) 2 nonalbino children with MN 

 blood type? 



(c) 3 children with M blood type? 



Table 1 



®Hm^m+<m + <m<$ 



Exact p= 0.4019 0.4019 



Number of "ones" = 1 



0.0321 0.0032 0.0001 



3 4 5 



